1
JEE Main 2023 (Online) 30th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are three non-zero vectors and $$\widehat n$$ is a unit vector perpendicular to $$\overrightarrow c $$ such that $$\overrightarrow a = \alpha \overrightarrow b - \widehat n,(\alpha \ne 0)$$ and $$\overrightarrow b \,.\overrightarrow c = 12$$, then $$\left| {\overrightarrow c \times (\overrightarrow a \times \overrightarrow b )} \right|$$ is equal to :

A
15
B
9
C
6
D
12
2
JEE Main 2023 (Online) 30th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let a unit vector $$\widehat{O P}$$ make angles $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes $$\mathrm{OX}$$, $$\mathrm{OY}, \mathrm{OZ}$$ respectively, where $$\beta \in\left(0, \frac{\pi}{2}\right)$$. If $$\widehat{\mathrm{OP}}$$ is perpendicular to the plane through points $$(1,2,3),(2,3,4)$$ and $$(1,5,7)$$, then which one of the following is true?

A
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
B
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
C
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
D
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
3
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \times \overrightarrow c = \overrightarrow 0 $$ and $$\overrightarrow r \,.\,\overrightarrow a = 0$$. Then $$\overrightarrow r \,.\,\overrightarrow c $$ is equal to :

A
36
B
30
C
34
D
32
4
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\overrightarrow a = 4\widehat i + 3\widehat j$$ and $$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightarrow b } \right) + 25 = 0,\overrightarrow c \,.(\widehat i + \widehat j + \widehat k) = 4$$, and projection of $$\overrightarrow c $$ on $$\overrightarrow a $$ is 1, then the projection of $$\overrightarrow c $$ on $$\overrightarrow b $$ equals :

A
$$\frac{3}{\sqrt2}$$
B
$$\frac{1}{\sqrt2}$$
C
$$\frac{1}{5}$$
D
$$\frac{5}{\sqrt2}$$
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